Abstract

We investigate the oscillations of slowly rotating
superfluid stars, taking into account the vortex mediated mutual friction force that
is expected to be the main damping mechanism in mature neutron star cores.
Working to linear order in the rotation of the star, we consider both the
fundamental f-modes and the inertial r-modes. In the case of the (polar) f-modes,
we work out an analytic approximation of the
mode which allows us to write down a closed expression for the mutual friction damping timescale.
The analytic result is in good agreement with previous numerical results obtained using an energy integral argument.
We extend previous work by considering the full range of permissible values for the vortex drag,
e.g. the friction between each individual vortex and the electron fluid. This leads to the first ever results for the f-mode
in the strong drag regime.
Our estimates provide useful insight into the dependence on, and relevance of, various equation
of state parameters. In the case of the (axial) r-modes, we confirm the existence of two classes of
modes. However, we demonstrate that only one of these sets remains purely axial in more realistic neutron star models.
Our analysis lays the foundation for companion studies of the mutual friction damping of the
r-modes at second order in the slow-rotation approximation, the first time evolutions for superfluid neutron star perturbations
and also the first detailed attempt at studying
the dynamics of superfluid neutron stars with both a relative rotation between the components and
mutual friction.

pacs:

Neutron stars have a complex interior structure. With core densities reaching several times the nuclear saturation density,
these objects require an understanding of physics that cannot be gained from laboratory experiments. This
makes the modelling of neutron star dynamics
an interesting challenge. On the one hand, one has to consider exotic physics that is, at best, poorly constrained.
On the other hand, one may ask to what extent observations can distinguish between
different possible models. An excellent example of this interplay concerns the possibility that the quarks may deconfine in the
high density region. If this is the case, it will have a considerable effect on transport properties associated with
viscosity and heat conductivity. In fact, such a quark core is expected to be a colour superconductor (1).
The dynamics of this exotic state of matter, and the relevance of its different possible phases, is not yet
certain. In order to improve our understanding of this problem, we need to build more precise stellar models and
study, for example, their oscillation properties in detail. In this context, considerable attention has been focused
on the inertial r-modes of a rotating star. The r-modes are interesting because they can be driven unstable by the
emission of gravitational radiation, see (2); (3) for literature reviews. The r-mode instability window is, however,
sensitive to the physics of the neutron star interior. Since the bulk and shear viscosities are
quite different in a quark core, compared to “normal” npe matter, one may hope to use observations to constrain the theory, see
(1) for a discussion of the relevant literature.
In absence of a direct detection of an r-mode gravitational-wave signal, this analysis would have to be
based on the nature of the instability window. The idea would be that, if an observed neutron star spins at a rate
that would place it inside a predicted instability region, one may be able to rule out this particular theoretical model.
Of course, this argument comes with a number of caveats. It could, for example, be that additional physics
places a stronger constraint on the r-modes than the considered mechanisms. Inevitably, this becomes a “work in progress” where
improved theoretical models are tested against better observational data.

In order to consider “realistic” neutron stars, it is important to appreciate the relevance of superfluidity.
A neutron star is expected to contain a number of superfluid/superconducting components (4), and it is
crucial to understand to what extent this affects the stars oscillation properties. It is well established that the
behaviour of a superfluid system can differ significantly from standard hydrodynamics. The most familiar low-temperature system is, perhaps,
He4, which exhibits superfluidity below a critical temperature near 2 K. Experimentally, it has been demonstrated that
this system is very well described by the Navier-Stokes equations above the critical temperature. Below the critical temperature
the behaviour is very different, and a “two-fluid” model is generally required (see (5) for a very recent discussion).
Superfluid neutron stars are, to some extent, similar. The second sound in Helium is analogous to a set of, more or less
distinct, “superfluid” oscillation modes (6); (7); (8) in a neutron star.
These additional modes arise because the different components of a superfluid system are allowed to move “through” each other.
The dissipation channels in a superfluid star are also quite different. Basically, the superfluid flows without friction.
In the outer core of a neutron star, which is dominated by npe matter, one expects the neutrons to be superfluid while
the protons form a superconductor. As a result, the shear viscosity is dominated by e-e scattering (9); (10).
The bulk viscosity, which is due to the fluid motion driving the system away from chemical equilibrium and the resultant
energy loss due to nuclear reactions, is also expected to be (exponentially) suppressed in a superfluid (4).
These effects have direct implications for the damping of neutron star oscillations, and play a key role in determining the
r-mode instability window for a mature neutron star. This is, however, not the end of the story. A superfluid exhibits an additional
dissipation mechanism, usually referred to as “mutual friction”. The mutual friction is due the presence of vortices in a rotating
superfluid. In a neutron star core, the electrons can scatter dissipatively off of the (local) magnetic field of each vortex (see (11); (12); (13)
for discussions and references). This effect may dominate the damping of realistic neutron star oscillation modes.

The basic requirements of a rudimentary model for superfluid neutron star oscillations should now be clear. One must account for the
additional dynamical degree(s) of freedom, and also account for the mutual friction damping. This is obviously only a starting point, but the
problem is sufficiently complicated that one may want to proceed with care.
There has already been a number of studies of
dissipative superfluid oscillations. The area was pioneered by Lindblom and Mendell who, in particular, demonstrated that the
gravitational-wave instability of the fundamental f-modes would be suppressed in a superfluid star (14). Following the
discovery of the r-mode instability, they also provided the first accurate estimates of the relevance of the mutual friction
for these modes (15). Similar results were subsequently obtained by Lee and Yoshida (16). These studies provide
important assessments of the relevance of the mutual friction damping.
There are, however, a number of reasons why we need to return to this problem. Most importantly, we want to consider more realistic
neutron star models, including finite temperature effects, magnetic fields and the possible presence of exotic (hyperon and/or quark)
cores. The additional physics brings additional complications, like additional fluid degrees of freedom, boundary layers at phase-transition interfaces
and fundamental issues concerning dissipative multifluid systems (5); (17). We also need to move away from the assumption that the vortex drag,
which leads to the mutual friction, is weak. Strong arguments suggest that this is not going to be the case when the protons form a type II superconductor
and there are magnetic fluxtubes present in the system (18); (19); (20). The neutron vortices may be “pinned” to the fluxtubes leading to a strong
drag regime. The strong drag problem has only been considered recently (21); (22), and the first results demonstrate the presence of a
new instability in systems where the two components rotate at different rates. This instability, which may be relevant for the understanding of pulsar glitches ((22)), provides
a direct demonstration that the dynamics in the strong drag regime may be both complicated and interesting.
The present investigation lays the foundation for future work in this direction by allowing for strong drag. In particular, we retain the dynamic
contribution to the mutual “friction” that has previously been neglected as a matter of course.

Our discussion is based on the standard two-fluid model for neutron star cores (23); (17). That is,
we consider two dynamical degrees of freedom loosely speaking representing the superfluid neutrons
(labeled n) and a charge-neutral conglomerate of protons and electrons (labeled p).
Assuming that the individual species are conserved, we have the
usual conservation laws for the mass densities ρx,

∂tρx+∇i(ρxvix)=0

(1)

where the constituent index x may be either p or n. Meanwhile, the equations of momentum balance can be written

(∂t+vjx∇j)(vxi+εxwyxi)+∇i(~μx+Φ)+εxwjyx∇ivxj=fxi/ρx

(2)

where the velocities are vix, the relative velocity is defined as wixy=vix−viy and
~μx=μx/mx represents the
chemical potential (we will assume that mp=mn throughout this paper). Φ represents the gravitational potential, and the parameter
εx encodes the non-dissipative entrainment coupling between the fluids (23); (17).
The force on the right-hand side
of (2) can be used to represent various other interactions, including dissipative terms (17).

In the following we will focus on the vortex-mediated mutual friction. Assuming that the two fluids exhibit solid body rotation
we have a force of form (13) (see also (11); (12))

fxi=2ρnB′ϵijkΩjwkxy+2ρnBϵijk^ΩjϵklmΩlwxym

(3)

Here, Ωj is the angular frequency of the neutron fluid (a hat represents a unit vector).
The mutual friction parameters are intimately related to the induced friction on the vortex. The latter is
often described in terms of a dimensionless “drag” parameter R such that

B′=RB=R21+R2

(4)

In the standard picture, the mutual friction is due to the scattering of electrons off of the array
of neutron vortices. This leads to R≪1, i.e., B′≪B, and hence the first term in the mutual
friction force can be ignored. There are, however, good arguments for why the problem may be in the opposite
regime. In particular if one considers the interaction between the fluxtubes in a type II proton superconductor
and the neutron vortices (18); (19); (20). Then one would expect to be in the strong drag regime
where R≫1, i.e., B′≈1 while B remains small. Superfluid oscillations in
this regime have not previously (with the
exception of (22)) been considered.

Anyway, from (3) we see that the mutual friction will not be present in a non-rotating star. This is obvious
since there would then be no vortices in the first place. Of course, any non-trivial motion of the
superfluid neutrons leads to vortex generation. This means that a generic perturbation of a non-rotating star will be associated with
a local vorticity which could lead to mutual friction. However, in this context the resulting
mutual friction interaction would require a perturbative calculation to be carried out to second order. As far as we are aware, such calculations have not yet been attempted.
It may be an interesting problem for the future.

iii.1 Decoupling the degrees of freedom

If we want to consider the effects of mutual friction
we need to consider rotating stars. To keep the problem tractable (at least initially) we
assume that the background configuration is such that the two fluids rotate together. Perturbing the equations of motion and working in a frame rotating with Ωj we then have

∂t(δvxi+εxδwyxi)+∇i(δ~μx+δΦ)+2ϵijkΩjδvkx=δ(fxi/ρx)

(5)

and

∂tδρx+∇j(ρxδvjx)=0

(6)

where δ represents an Eulerian variation.

From previous work on superfluid neutron star perturbations (and indeed the large body of work on superfluid Helium)
we know that the problem has two ”natural” degrees of freedom, see for example (6); (7); (8); (24); (25); (26). One of the degrees of freedom
represents the total mass flux.
Introducing

ρδvj=ρnδvjn+ρpδvjp

(7)

and combining the two Euler equations we find that

∂tδvi+∇iδΦ+1ρ∇iδp−1ρ2δρ∇ip+2ϵijkΩjδvk=0

(8)

where ρ=ρn+ρp and the pressure is obtained from 1

∇ip=ρn∇i~μn+ρp∇i~μp

(9)

In deriving this relation we have used

ρn∇iδ~μn+ρp∇iδ~μp=∇iδp−δρ∇i~μ=∇iδp−1ρδρ∇ip

(10)

where it has been assumed that the two fluids are in chemical equilibrium in the background. That is, we have
~μn=~μp=~μ.
We also have the usual continuity equation

∂tδρ+∇j(ρδvj)=0

(11)

At this point we have two equations which are identical to the perturbation equations for a single fluid
system. It is particularly notable that (8) does not have a force term. This follows
immediately from the fact that we are only considering the mutual friction interaction. In other situations, say
including shear viscosity, we would no longer have a homogeneous equation.

Of course, we are considering a two-fluid problem and there is a second degree of freedom to take into account.
To describe this, it is natural to consider the difference in velocity. Thus, we introduce

δwj=δvjp−δvjn

(12)

Combining the two Euler equations in the relevant way we have

(1−¯ε)∂tδwi+∇iδβ+2¯B′ϵijkΩjδwk−¯Bϵijk^ΩjϵklmΩlδwm=0

(13)

Here we have defined

δβ=δ~μp−δ~μn

(14)

which represents the (local) deviation from chemical equilibrium induced by the perturbations. We have also introduced
the simplifying notation

¯ε=εn/xp,¯B′=1−B′/xp,¯B=B/xp

(15)

where xp=ρp/ρ is the proton fraction.
Again, equation (13) does not couple the different degrees of freedom.

The coupling is entirely due to the second continuity equation. It is
natural to use the proton fraction to complement the total density ρ. Then we find that

∂tδxp+1ρ∇j[xp(1−xp)ρδwj]+δvj∇jxp=0

(16)

This equation shows that the two dynamical degrees of freedom are explicitly coupled
unless the proton fraction is constant. This fact has already been
pointed out by Prix and Rieutord (24).

Before moving on, it is worth asking to what extent it is possible to find solutions that are
purely co-moving, i.e. for which δwj=δβ=0. From the above equations it is easy to see that
such a solution would have to satisfy

∂tδxp+δvj∇jxp=0

(17)

This condition is trivially satisfied if the proton fraction is uniform. In addition, it will be satisfied for fluid motion
that has (for a spherical background configuration) no radial component and also do not lead to variations in δxp.
Are there oscillation modes with this character? Indeed, to leading order in the slow-rotation approximation the standard r-mode
satisfies these criteria. It is purely axial and the associated density perturbations appear at order Ω2. However, in general
we do not expect to find any oscillations of a “realistic” neutron star model to be purely co-moving. This means that a generic
neutron star oscillation mode will be affected by mutual friction.

iii.2 Boundary conditions

To completely specify the perturbation problem, we need boundary conditions. At the centre of the star
we simply require that all variables are regular. The surface of the star is somewhat more complex.
In reality one
does not expect the superfluid region to extend all the way to the surface. A real neutron star will
always be covered by a single fluid envelope (eg. the outer parts of the elastic crust). However, for simplicity
we do not want to deal with the various interfaces in the present analysis (15); (27); (28).
Instead we will consider stars with a two-fluid surface, which is obviously somewhat artificial.

A reasonable approach is to assume that the perturbed star has a unique surface. That is,
let the two perturbed fluids move together (in the radial direction) at the surface.
In the two-fluid problem we have two distinct Lagrangian displacements ξjx(29).
These follow from

∂tξix=δvix+ξjx∇jvix+vjx∇jξix

(18)

We have assumed that the two fluids corotate in the background, i.e. we have vin=vip.
If we also impose that there is a common surface, then we have ξrn=ξrp at r=R
and it follows that we should require

δwr=δvrp−δvrn=0, at r=R

(19)

From (13) we see that this implies that, for a non-rotating configuration we must also have

∂rδβ=0, at r=R

(20)

When we determine the rotational corrections to the f-mode we will impose this condition also at first
slow-rotation order. This is not entirely consistent, cf. (13), but it is straightforward to relax this condition
should it be required.

If the two fluids move together at the surface, it also follows that

δp+ρξj∇j~μ=δp+ξr∂rp=Δp=0, at r=R

(21)

where ξr=ξrn=ξrp at the surface. This is the usual single fluid
condition of a vanishing Lagrangian pressure variation Δp.

iii.3 A bit of chemistry

Consider the various equations that we have written down. At this point the two
degrees of freedom [δvi,δp] and [δwi,δβ] only couple explicitly through
(16). In fact, if we assume that the two fluids are incompressible,
then there is no coupling at all. Since the mutual friction only enters the problem via (13),
it is thus the case that any incompressible dynamics in the [δvi,δp] sector will be unaffected
by mutual friction. This shows that, if we are interested in the effect of mutual friction on (say) the f-mode
oscillations of a star it is not meaningful to consider an incompressible model. We know already from the outset that we
would only find the mutual friction effects on the counter-moving “superfluid” modes. This may be an interesting
problem, but it is not our main motivation here.

For compressible models, the two degrees of freedom also couple indirectly. Basically, we need to use the equation
of state to relate [δp,δβ] to [δρ,δxp]. For models where the two
fluids co-rotate in the background we can use 2

δρ=(∂ρ∂p)βδp+(∂ρ∂β)pδβ

(22)

and

δxp=(∂xp∂p)βδp+(∂xp∂β)pδβ

(23)

Using these relations, or their “inverse”, we see that the two degrees of freedom couple in a more subtle way.
If we choose to reduce the problem by eliminating δp and δβ then the coupling arises
through the boundary conditions and the Euler equations.
If, on the other hand, we eliminate δρ and δxp then
the coupling enters through the continuity equations.

In order to estimate the damping associated with various dissipation mechanisms one can either (i) account for the
dissipative terms in the equations of motion and solve for the damped modes directly, or (ii) solve the
non-dissipative problem and use an energy integral argument to estimate the damping rate. In the typical situation when the damping
is very slow the second strategy should be reliable. Indeed, all studies of damped neutron star oscillations have
used this approach (see (3) for a discussion). Given this, it is natural to pause and consider the energy integral
approach to the mutual friction problem.

iv.1 The conserved energy

To work out a suitable energy associated with a given perturbation, we first multiply (5) with
ρxδ¯vix (where the bar represents complex conjugation). Then we add the result to its
complex conjugate. Combining the individual contributions from the neutron and proton fluids
and integrating over the star we
find that, when fxi=0, the result is a total time derivative of two terms. The first term is the
“kinetic energy”, which follows from

Ek=12∫[(ρn−2α)|δvn|2+4αRe(δ¯vinδvpi)+(ρp−2α)|δvp|2]dV

(24)

where 2α=ρxεx.
Alternatively, expressing this in the co- and countermoving variables, we have

Ek=12∫ρ[|δv|2+(1−¯ε)xp(1−xp)|δw|2]dV

(25)

The “potential” energy requires a bit more work. Using the divergence theorem and the continuity equation one can show that we need

(c.c. represents the complex conjugate). The surface term vanishes if ρx→0 at the surface of the star.
It also vanishes for modes that have no radial component. Adding the contributions for the neutron and proton fluids we see that we need

With these definitions it follows that the total “energy” is conserved, i.e.

∂tE=∂t(Ek+Ep)=0

(31)

when fxi=0. These energy expressions are equivalent to those used by Lindblom and Mendell (14).

iv.2 Mutual friction

Even though we will include the mutual friction terms in the equations
of motion, it is useful to work out the corresponding dissipation integrals.
After all, this is the way that the mutual friction damping has
traditionally been evaluated (14); (15); (16) and we want to be able to compare the
two approaches.

First consider the B′ terms. It is easy to show that
these terms are not dissipative.
We find that

2∂tEB′=2∫B′ϵijkΩj[δ¯vinδwknp+δ¯vipδwkpn+c.c.]dV=0

(32)

by symmetry. This result is not surprising. In fact, we see from (13) that the B′ terms enter the equations of motion in the same way as the Coriolis force.
Since the Coriolis terms vanish identically when we multiply each Euler equation with δvix this should be true also for the non-dissipative part of the mutual friction.

Finally, it is straightforward to show that the dissipative terms lead to

Let us now ask how we can use these results to estimate the mutual friction damping timescale. Let us assume that we have a mode solution to the full dissipative problem.
That is, we have a solution with time dependence eiωt where ω=ωr+i/τ such that τ is the damping timescale. From the fact that the energy is
quadratic in the perturbations it follows that (35)

τ=∣∣∣2E∂tE∣∣∣

(34)

Moreover, since the solution satisfies the dissipative equations of motion we also know that

∂tE=∂tEB

(35)

Hence, we can equally well use

τ=∣∣∣2E∂tEB∣∣∣

(36)

As long as we are using the complete solution to evaluate this expression, it is an identity. However, in many cases we do not
have access to the solution to the dissipative problem. (If we did, we would not need the energy integrals in the first place.)
In these cases we can still estimate the damping timescale by evaluating the right-hand side of (36) using the non-dissipative mode solution.
When the damping is sufficiently slow, in the sense that the dissipative terms have a small effect on the eigenfunctions, this estimate should be
reliable. Of course, one should not expect it to yield exactly the same result as the solution to the full dissipative problem.

iv.3 Gravitational-wave emission

Finally, let us work out the multipole formulas for
gravitational-wave emission from a two-fluid star.
This exercise is particularly relevant if we are interested in oscillations
that may be driven unstable by gravitational-wave emission (3).
The main motivation for including it here is that it demonstrates
the intuitive result that gravitational waves are only
generated by the co-moving degree of freedom.

Let us now return to the problem of oscillating superfluid neutron stars. We will first
derive the general perturbation equations for a slowly rotating superfluid star.
To do this we expand all variables in spherical harmonics. Since we expect rotation to couple the various multipoles,
we represent the velocity perturbations by the general expressions

Note that we represent the “co-moving” degree of freedom by the uppercase amplitudes [Wl,Vl,Ul] while the “counter-moving”
degree of freedom corresponds to the lowercase quantities [wl,vl,ul].
All scalar perturbations are expanded in spherical harmonics, i.e. we have δp=∑lδplYml etcetera.
From now on the sum over l will be implied.

One can use a number of different strategies in writing down the perturbation equations. To some extent this is a matter
of taste. However, in the slow-rotation problem it can be advantageous to work with a set of equations where the
coupling between different multipoles is minimal. The set of equations that we use was chosen using this criterion.
We also decided to use the velocity perturbations as our main variables. This approach is analogous to that used by Lockitch
and Friedman (32) in their analysis of inertial modes of single fluid stars. It is notably different from the two-potential formalism pioneered
by Ipser and Lindblom (33), which was extended to superfluid stars by Lindblom and Mendell (15).

We replace each of the perturbed Euler equations with three equations. The first is the radial component of the vorticity equation that
follows if we take the curl of (8) or (13). Assuming that the perturbations have a harmonic dependence on time, exp(iωt), we get

In deriving these equation we have made use of the standard recurrence relations

cosθYml=Ql+1Yml+1+QlYml−1

(57)

and

sinθ∂θYml=lQl+1Yml+1−(l+1)QlYml−1

(58)

where

Ql=[(l−m)(l+m)(2l−1)(2l+1)]1/2

(59)

For future reference, note that Qm=0 and Q2m+1=1/(2m+3).

Next we could use also the θ (or φ) components of the
vorticity equation. However, as discussed in (34)
there is a slightly simpler alternative. We first of all use a pair of equations analogous to the “divergence” equation in (34).
These can be written

This completes the description of the general first order slow-rotation problem.

In order to deduce the relevant recurrence relations from the above equations
we need to recall that we have been implying summation over l. That is,
we are considering relations of form

∑l[alQlQl−1Yml−2+blQlYml−1+clYml+dlQl+1Yml+1+elQl+1Ql+2Yml+2]=0

(66)

Using orthogonality of the spherical harmonics, i.e. multiplying by
¯Ymn and integrating over the sphere, we obtain the recurrence
relation

an+2Qn+1Qn+2+bn+1Qn+1+cn+dn−1Qn−1+en−2Qn−1Qn=0

(67)

Given this result, it is straightforward to write down recurrence relations
for the various classes of oscillation modes of a rotating superfluid star.
However, since the level of rotational coupling is different for different kinds of modes,
it is not particularly useful to write down the general relations.
Instead, we focus on two specific examples.

Let us begin by considering modes that are non-trivial already in a non-rotating star.
Then we first need to solve the non-rotating (and non-dissipative since the
mutual friction damping requires rotation) problem. Simply setting
Ω=0 in our perturbation equations we see that the polar and axial degrees of
freedom decouple (as they should). It is also clear, cf.
(55) and (56), that there will not exist
any purely axial modes in the non-rotating case.
This means that we can make the Ansatz

ω=ω0+ω1Ω

(68)

together with

Wl=W0l+ΩW1l,Vl=V0l+ΩV1l,Ul=ΩU1l

(69)

and

wl=w0l+Ωw1l,vl=v0l+Ωv1l,ul=Ωu1l

(70)

and similarly for the various scalar perturbation quantities. For example, in the case of the proton fraction we
have δxp=∑lδxlYml with

δxl=δx0l+Ωδx1l

(71)

vi.1 The non-rotating problem

At the non-rotating level the equations in Section IV provide the following relations

ρnδ~μ0n,l+ρpδ~μ0p,l=−iω0ρV0l

(72)

ρnr∂rδ~μ0n,l+ρpr∂rδ~μ0p,l=−iω0ρW0l

(73)

δβ0l=−iω0(1−¯ε)v0l

(74)

r∂rδβ0l=−iω0(1−¯ε)w0l

(75)

Meanwhile the continuity equations lead to

iω0r2δρ0l+∂r(rρW0l)−l(l+1)ρV0l=0

(76)

and

iω0ρr2δx0l+∂r[xp(1−xp)rρw0l]−xp(1−xp)l(l+1)ρv0l+ρW0lr∂rxp=0

(77)

Before we proceed, we will simplify the problem. Our aim is to determine analytic approximations for the fundamental
modes of the system, including the mutual friction damping. Solving the problem numerically is, of course, straightforward but
does not lead to the same level of insight into the dependence on the various parameters.
To facilitate an analytic solution, we will combine an incompressible background model
with compressible perturbations. This simplifies the calculations considerably.
In addition, since this is the same model that was considered by Lindblom and Mendell (14) we can compare our final results directly
to the available literature.
We thus assume that
ρn and ρp are both constant, while δρn and δρp are not.

It is also useful to introduce
a new variable for the co-moving degree of freedom. Let us define

δhl=1ρδpl=1ρ(ρnδ~μln+ρpδ~μlp)

(78)

For a single barotropic fluid, δhl corresponds to the perturbed enthalpy. For a compressible background model
we would have

ρnr∂rδ~μln+ρpr∂rδ~μlp=ρr∂rδhl−ρδβlr∂rxp

(79)

However, for the uniform density model the gradient of the proton fraction vanishes so we simply have

Before we proceed, we need to decide what variables we want to work with.
We can either remove [δhl,δβl] or
[δρl,δxlp] (or some other combination of these variables) from the problem using thermodynamic identities. Opting for the latter
possibility, we use